Hello! I don't have access to the doc you linked so I can't view it. To diagnose the problem, it would be helpful if you showed the error message(s) as well and described what exactly you mean by "struggles to show custom dialogue" since "struggles" isn't exactly a coding term. I'm also not sure what you mean by your second note about the custom pronoun set selection - please walk me through the steps you are taking, what you see, and what you expect to see.
So, these are pretty standard errors with your code rather than the plugin.
elif pronoun == "custom" and "pronoun": makes no sense as a condition, for example; I don't really know what you were trying to write here, but this line will always be truthy if the previous check isn't because and "pronoun" is always truthy. You might want to look at https://feniksdev.com/conditional-statements-in-renpy/ for more on how to construct conditional statements.
You also have a lone elif : with no condition; you probably meant to write else:
I'm not entirely sure what your issue is with the custom pronouns where you mention "Appears after trying to use both one and custom set of pronouns, doesn’t show the singular pronoun menu after using entering custom pronouns" - that's how that code works. It's working as it's written and as was intended. You input custom pronouns and pick your terms and that's it. It's not meant for picking multiple pronoun sets; there's a different example specifically for that.
You also have a screenshot of the example pronoun adjustment screen with the note "Showing screen for multiple pronouns shows up as this" but it looks exactly as I would expect, as far as I can tell. Feel free to adjust the styling however you want if the size isn't suitable for you.
Your LabelNotFound error states exactly what's wrong - you have a jump to multiple_pronouns_demo which I presume you intend to change to jump somewhere else since you don't have or perhaps need a demo label.
You also note that the dialogue later is incorrect since you have custom and they/them pronouns, but your confirminfo label doesn't take any of that into account. It just prints the current pronoun set. You'll need to use something like "Your pronouns are [pretty_print_pronouns()]." if you want it to print out a list of all the player's pronouns rather than just their current pronouns.
I hope that helps you get started fixing things!